254 



REPORT 1865. 



a for the side plates AD, BD 1 down to the top of bilge, d being their depth ; 

 and so on, as before ; the equations (1) and (3), art. 18, 19, will become 



-io„ 



l =Qo+2j a ^- a J 2 = 1 Vo^+2:J « ! (7 i - a J 2 



(1) 



(2) 



where Q is the moment of inertia of the side plates about their centre of 

 gravity. It will be observed that (Ji — a t J' is always positive. 



And we further have 



8 



M=|l 



h 



and 



' r ^o 



S= 



m 



R ma 2 



O, as — f — . 



(3) 



W 



Fiar. 14. 



If the plates AE, ED have not the same thickness, then Q, of formula (3), 

 art. 19, must be retained, and Q and Q x calculated accordingly. 



22. If a ship is supported at its extremities, and the load distributed as de- 

 scribed in art. 10, then equations (6) or (2), accord- 

 ing to circumstances, will give the value of II in the 

 foregoing formulas ; or if the ship be supported at 

 the centre, then equations (10) or (11), art. 12, will 

 give the value of M, and so on to other cases. 



To adopt a more summary though less accurate 

 method of calculation, let AI) represent the section 

 of the material of an iron ship ; the flange «, the 

 section of the materials of the main deck ; the 

 flange a. the section of the lower deck ; the flange 

 a the section of the bottom ; a the section of the 

 material in the side plates or rib, d being the 

 depth, and so on, as before. In this case equations 

 (1) and (2), art. 21, become 



I 



h=^(a u a + a l a. L + a.jt 2 + a 3 a 3 ), 



(5) 



l^^a^l + a.ih-ccJ + a^h-ccJ' + aXh-aJ + a.Xh-cc.Y (6) 



If we take d =~D, the whole depth of the section, « 2 = D, « =^ D, and 

 a = 0, these formulas become 



h=l^a + a 3 )J) + a 2a2 ], (7) 



Io=«o{-iVD 2 + ^-P) 2 } + ^ 2 + ^-« 2 ) a + ^^-D) 2 (8) 



VTJien the depth and the amount of material in the transverse section are 

 given, to determine the distribution of the material so that the beam may have 

 the greatest strength. 



23. The first and most important condition of maximum strength is, that 

 the material should be accumulated, as far as practicable, at the upper and 

 lower edges of the beam. 



In tubular girders, where the material is accumulated at the upper and 

 lower edges of the girder, the sectional areas at these parts should be in the 

 inverse ratio of the ultimate resistances of the material in these parts re-» 

 spectively. 



